Regularisation

Ch. 4 considers the regularisation of some basic divergent series. The first is the geometric series, which is shown to be conditionally convergent outside the unit circle of absolute convergence for Rz<1 and divergent elsewhere. Nevertheless, the regularised value is found to be identical to the limit of 1/(1−z)when the series is convergent. Then the regularisation of the binomial series is considered, where again, the regularised value is found to equal the limit when the series is convergent. Next the series denoted by 2F1 (a+1,b+1;a+b+2−x; 1), which is divergent for Rx>0, is analysed. Here the regularised value is found to be different from the limit when the series is convergent or for Rx<0. Finally, the harmonic series is studied, whose regularised value equals Euler’s constant. Unlike the previous examples, the last example involves logarithmic regularisation.